All Questions
Tagged with expected-value regression
69
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74
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Expectation of residuals in linear regression
Consider a linear regression model
$$ Y=X\beta + \epsilon, $$
where $Y\in R^n$, $X = (x_1,...,x_n)^T\in R^{n\times p}$ are i.i.d. $p$-dimensional observations, $\beta\in R^p$, and $\epsilon = (\...
- 79
2
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2
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90
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Why is every predicted $y_i$ in linear regression equal to $E[Y|X]$?
I have some loose intuition with this but I don't understand it
Say we have a scatterplot of data in 2 dimensions. Then we can propose a mean model where, for all $x_i \in X$ the estimated $y_i = E[Y]$...
- 127
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19
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Find function $f$ such that the performance of models $h_1$ and $h_2$ is similar
Suppose that $h_1$ and $h_2$ are the first and third order polynomials respectively, which are obtained by solving the OLS equation (using the training dataset). Also consider the following ...
- 67
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11
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Calculating Best Linear Predictor: vector transposed translated into non-transposed vector?
I am studying the best linear predictor part of Conditional Expectation and the Projection part about the regression model.
while we are looking for the Beta that minimizes S(B), I quite do not ...
- 1
2
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2
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166
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How should I interpret the assumption of the regression?
I read an econometrics book which states one of the basic assumptions of regression is that
$$E(u|x) = 0$$
In another book however I see it written that
$$E(u_i|x_i) = 0$$
Are these two saying the ...
1
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0
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40
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Intercept change issues in growth curve model in R
I am currently analyzing data using a growth curve model, and I have one question that puzzles me. I wonder why the intercept values keep changing when I add random effects or higher-order terms to ...
- 11
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52
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Questions about the expected target value $\bar{y}(x)$
Assume a simple Linear Regression problem, where we have $n$ data points $x$, and one target variable $y$.
My confusion, or more precisely misconceptions start at the following equation and its ...
- 535
4
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1
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59
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Finding most probable vector, given angles
Assuming you have a series of known vectors $k_i$ in $\mathbb{R}^N$, each at an angle $\theta_i$ to a single unknown vector $v$. All vectors have the same dimension $N$ which can be arbitrarily ...
- 73
1
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1
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142
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In an RCT, does running OLS on $Y_i = \beta_0 + \tau D_i + \varepsilon_i$ and recovering $\tau$ recover ATE or ATT
Let's say I run an RCT and then run OLS on $Y_i = \beta_0 + \tau D_i + \varepsilon_i$ where $D_i$ is a dummy variable indicating whether an individual $i$ received the treatment. If I were to take the ...
- 11
1
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1
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88
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Finding the maximum likelihood solution corresponds to finding the root of a regression function. How?
Given a pair of RVs $z,\theta$ governed by a joint distribution $p(z,\theta)$. Conditional expectation of $z$ given $\theta$ defines a deterministic function (called as regression functions) $f(\theta)...
- 93
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2
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1k
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Find the expected value for the sum of squares of regression
For a multilinear regression model, I'm trying to find the expected value of the sum of squares of regression (SSR). I have so far,
$$E(SSR) = E(\hat y'\hat y) = E((X\hat\beta)'(X\hat\beta)) = E((X(X'...
- 31
1
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1
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759
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Expectation of MLE estimator for $\sigma^{2}$ Multiple Regression model
Given Y = $X\beta + \epsilon$, where $\epsilon \sim N(0,\sigma^{2}I_n)$,
Under this setting, I get the MLE estimator for $\sigma^{2} = \frac{1}{n} ||Y - HY||^{2}$, where H is the hat matrix $X(X^TX)^{-...
1
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0
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150
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"Linearity" of the Normal Distribution
I am trying to understand the following statement:
Can someone please explain what is meant by "the conditional expectation function m(x) is linear in x"?
In the case of regression, I ...
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42
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relationship between expectation of a ratio of random variables and a ratio of expectation of random variables [duplicate]
I know that in general for random variables $X$ and a function $g$
$E[g(X)]\ne g[E(X)]$.
I would like to know when the following holds
$E_Z\Big[\frac{cov(X,Y|Z)}{var(X|Z)}\Big] = \frac{E_Z[cov(X,Y|Z)]}...
- 236
3
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1
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323
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How to interpret mean of this estimated AR(1) process
I estimated an AR(1) process, my data looks like this:
Making usual unit root test, they suggest that an estimated AR(1) from this data is stationary. Estimating the AR(1) over this data, these are ...
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